Physics - Electrostatics Question with Solution | TestHub

When a conducting sphere is grounded, its electrostatic potential becomes zero. Let $q_{ind}$ be the charge induced on the sphere. The potential at the center of the sphere (or any point on its surface) due to the external point charge $Q$ and the induced charge $q_{ind}$ must sum to zero. The potential at the center due to the external charge $Q$ is $V_Q=k\frac{Q}{2R}$, where $k=\frac{1}{4\pi {ϵ}_0}$. The potential at the center due to the induced charge $q_{ind}$ (which effectively acts as if it's at the center for potential calculation at the center) is $V_{q_{ind}}=k\frac{q_{ind}}{R}$. Setting the total potential to zero: $V_Q+V_{q_{ind}}=0⟹k\frac{Q}{2R}+k\frac{q_{ind}}{R}=0$. Dividing by $k/R$, we get $\frac{Q}{2}+q_{ind}=0$. Therefore, the induced charge on the sphere is $q_{ind}=-Q/2$.